## Abstract

Flexibility is a crucial component of competitive advantage, especially under conditions of dynamically changing environments. In this theoretical paper, we present alternative flexibility measures that can be used to calculate the ability of the production technology to accommodate output variations at lower costs. We refine the existing short-run multi-output flexibility measure to the long-run using three alternative total cost functions in order to capture the impact of fixed costs on adjustment ability. The proposed new long-run flexibility measures are based on alternative definitions of the long-run total cost function considering various assumptions about the decision-making process. We derive primal flexibility measures by using the dual relationships between these cost functions and the corresponding input distance functions. The proposed measures allow us to investigate both short-run and long-run flexibility using different representations of the production technology, and thus to examine various aspects of the firm’s ability to adjust.

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## Notes

There are, of course, some other definitions of flexibility in economic literature. Some authors suggest using the second derivative of the total cost function (cf. Tisdell 1968; Mills 1984; Zeller and Robison 1992; Zimmermann 1995), whereas other authors propose supply elasticity (Mills and Schumann 1985; Das et al. 1993) as a measure for flexibility. These measures are formulated for the single-product firms only. We do not consider them in our paper and concentrate on the more general case.

This function is sometimes called the restricted cost function.

We adopt the convention that an

*n*-vector is a column vector \(\left( {n \times 1} \right)\) and that \({\mathbf{w'x}}\) is the inner product, with “\({\mathbf{'}}\)” being the transpose operator. For the Jacobian and the Hessian, we first derive “vertically”, that is, if we have a function \(f\left( {{\mathbf{x,y}}} \right)\) with**x**as an*n*-vector and**y**as an*s*-vector, then the vector of partial derivatives with respect to**x**is denoted**f**_{ x }and its dimension is \(\left( {n \times 1} \right)\). If we derive once again with respect to**y**, we derive “horizontally”, that is,**f**_{ xy }is a \(\left( {n \times s} \right)\) matrix. Vectors and matrices are represented using non-italic, bold characters. Scalars are in italic, non-bold characters. The first-order derivatives of an*n*-column vector, say the conditional factor demands**x**(**w**,**k**,**y**), with respect to an*s*-vector, say the outputs, is a matrix \(\left( {n \times s} \right)\).In this paper, we are interested in the relation between cost and the output. For that reason, the properties of the cost function with respect to the input prices (homogeneity, concavity and monotonicity) do not play any role.

Relaxing those assumptions makes the dynamic problem non-autonomous and the Hamilton–Jacobi equation includes additional terms, which makes the problem very complex. Furthermore, the introduction of expectations makes the problem almost intractable. See Lasserre and Ouellette (1999) on this point.

It is possible to define the adjustment cost in terms of net investment (\({\dot{\mathbf{k}}}\)

**)**instead of gross investment (**I)**.For this explanation, it is not important to specify which cost function is under investigation. For that reason, we use

*C*as the representative cost function.It should be clear that we could choose a different path in the output space using a directional approach. As mentioned in Crémieux et al. (2005), it is possible to replace τ with

*α*_{ i }τ, where*α*_{ i }represents the non-proportional increase in the*i*th output. Obviously, the resulting measure would be different, as the outputs would not be the same along the other directions. It must also be clear that the cost function in itself is not affected by the choice of a path in the output space. In particular, we do not make any assumption on returns to scale or scope economies or homotheticity.We are indebted to a referee for this suggestion.

Here, the notation is a bit awkward. The typical (

*a*,*b*) element of the (*s*,*s*)-square matrix \({\mathbf{VC}}_{{{\mathbf{yyk}}}} {\mathbf{k}} = \frac{{\partial^{2} {\mathbf{VC}}_{{\mathbf{k}}}^{\prime} {\mathbf{k}}}}{{\partial {\mathbf{y}}\partial {\mathbf{y}}{^{\prime}}}}\) should be read as \(\left[ {\sum\nolimits_{i}^{m} {\frac{{\partial^{3} VC}}{{\partial y_{a} \partial y_{b} \partial k_{i} }}k_{i} } } \right]\). It would be more appropriate to use the*vec*operator to represent third-order derivatives. In that case, it is necessary to vectorize the Hessian**VC**_{ yk }and then to derive with respect to**y**. We feel that the reading of the resulting matrices would be less intuitive. Since**k**does not depend on**y**and**VC**_{ k }**’k**is a scalar in this case, our convention leads to a simpler equation. It must be clear that \({\mathbf{y}}{^{\prime}}\left( {{\mathbf{VC}}_{{{\mathbf{yyk}}}}^{\prime} {\mathbf{k}}} \right){\mathbf{y}}\) is a scalar.Crémieux et al. (2005) proposed this measure and applied it to calculate long-run flexibility in the Québec hospital sector.

The short-run variable cost function as an intermediate step of the long-run dynamic decision process is expressed by \(VC\left( {{\mathbf{w}},{\mathbf{k}},{\mathbf{I}},{\mathbf{y}}} \right) = \mathop { \hbox{min} }\limits_{{{\mathbf{x}} > 0}} \left\{ {{\mathbf{w'x}}|D^{S} \left( {{\mathbf{x}},{\mathbf{k}},{\mathbf{I}},{\mathbf{y}}} \right) \ge 1} \right\}\).

The dual representation for

**VC**_{ kk }can be derived after differentiating the same set of the equations with respect to**k**. The solution is analogous to the one presented in this paper.This primal short-run flexibility measure was proposed by Renner et al. (2013).

Dual relationships between long-run total cost function and long-run input distance function are derived analogous to the case for the variable cost function. It suffices to mimic the proof.

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Renner, S., Glauben, T., Hockmann, H. *et al.* Primal and dual multi-output flexibility measures.
*J Prod Anal* **44**, 127–136 (2015). https://doi.org/10.1007/s11123-015-0449-8

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DOI: https://doi.org/10.1007/s11123-015-0449-8